Now suppose the probability of treatment changes

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Unformatted text preview: ntive justification. Now suppose the probability of treatment changes discontinuously at a threshold,
but not from 0 to 1. This is a situation for applying FRD. The “fuzziness” in FRD comes from the change in probability of treatment, not
fuzziness about the threshold For example, decision to offer a scholarship based on: Continuous measure of academic ability (e.g., GRE) exceeds given cut-off, and Subjective information (e.g., recommendation letters) observed only by the
evaluator FRD is like a random experiment with non-compliers no-shows: treatment group members who do not receive treatment, and cross-overs: control group members who do receive treatment In sharp RD designs, the jump in the outcome at the cutoff is the estimate of the
causal impact of the treatment Fortin – Econ 560 Lecture 0 In a FRD design, the jump in the outcome is divided by the jump in the probability
of treatment at the cutoff to produce a local Wald estimate (equivalent to a local IV
estimate) of the causal impact
lim ↓ [ | = ] − lim ↑ [ | = ]
lim ↓ [ | = ] − lim ↑ [ | = ]
= [ (1) − (0)|unit i is a complier and = ]
= where “compliers” are units that receive the treatment when they satisfy the cutoff
rule ( ≥ ), but would not otherwise receive it. The effect of interest as the ratio of the “reduced form” discontinuity in the relation
between Y and X to the “first-stage” discontinuity in the relation between D and X. The important point to remember is that fuzzy RD is numerically equivalent (and
conceptually similar) to IV and can also be estimated by 2SLS. An additional graph is needed when doing FRD: probability of treatment by
assignment variable should show a discontinuous probability at the threshold. Fortin – Econ 560 Lecture 0 There are three general types of threats to an RD design
1. Other variables change discontinuously at the cutoff For example, 18th birthday marks a discontinuous change in eligibility to vote,
but also eligibility for draft, sentencing as an adult, and lots of other things,
which may or may not be relevant depending on the outcome in question Test for jumps in covariates, including pret...
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